1) As Phobos said — that's APD's MO.
2) As far as I've seen, we either get plothole or "answered an ad" (Rick, Techno-Dann, Agent!Desdendelle, etc).
3) As long as it isn't stated otherwise, you can join an RP without having Permission.

You are looking for the All-Purpose Department. Agents specialize in a single continuum and do all types of missions in that one world.

-Phobos

1) Hmm... DMS has divisions for all continuums desired, but I don't believe multi-departments are a thing. Your best option would be Floaters.

2) Multiple, multiple ways... which oddly enough usually involves falling through a plothole. But others (like Rick) simply answered a shady advertisement.

3) Hm... somebody else answers the RP question, please.

Making a Character Replacement of at least two characters, lack of proper dialogue punctuation, calling Finnick "Finnie", making Prim a complete jerk, etc, etc. I haven't even looked that much at it.

The shortest badfic ever sporked (this — NSFW & NSFB!) clocks at 138 words, which are a couple of sentences. Usually, stuff that's too short gets passed over because there's not enough material to actually spork. That amount, however, changes from person to person.
Looking at the fic linked above, it is a bit less than 6,000 words; that's plenty for a shortish mission &dmash; that's hS' cutoff for Driftwood, and Driftwood makes for an excellent example of a PPC... spinoff? series? in my opinion.
So yeah, that fic is long enough.